2) Analytical approach: Rosensweig theory (LRT)
Solving the master equation in the linear regime:
Energy loss per cycle:
Imaginary susceptibility:
Equilibrium susceptibility:
3) Computational methods: kinetic Monte-Carlo model
● Individual particles are described by the Stoner-Wohlfarth theory:
● distributions of particle volumes;
● distributions of particle anisotropy value;
● random distributions of uniaxial anisotropy vectors;
● Thermal bath is included in the models, allowing capturing both superparamagnetic and hysteretic regimes.
● Inter-particle interaction are modelled as dipole-dipole interactions.
● Various spatial arrangements of nanoparticles can be considered.
4) High efficiency in transition region between RT and fully hysteretic regimes System of spherical nanoparticles with:
● Random spherical distribution of easy axis
● Log-normal distribution of size ( σd=0.1) and anisotropy (σ
K=0.1)
5) Role of interactions
1) Motivation
Magnetic hyperthermia is a promising methodology
for cancer treatment.
Clinical requisites:
Accurate ΔT: Ttreatment ~ 42º - 45ºC
Biocompatibility (composition; coating; dose)
Size ~ 10 -100 nm
Limited HAC(f*Hmax
< 6*107 Oe/s): Hmax~[5-200] Oe; f~[0.1-1] MHz
Unified model of hyperthermia via hysteresis heating in systems of interacting magnetic nanoparticles
Sergiu Ruta1, Ondrej Hovorka2, Roy Chantrell1
1 Physics Department, University of York, York, UK2 Faculty of Engineering and the Environment, U. of Southampton, Southampton, UK
ΔU=Hmax2 2π f χ
' '
χ' '=χ0
1+(2π f τeff )22π f τeff
τN=1
2 f 0
eKVK bT
χ0=[ M s L(α)
H ]H=0
α=M sV H
K bT
Analytical approach:
● Rosensweig theory (RT)
Simulations:
● Minor hysteresis cycle
?
H=300 Oef= 100 KHz
Size and anisotropy are optimised
0.9 0.8 0.75
For SAR>0.9 of max SARD: >5nm tolerance
For SAR: >0.9 of max SARD: <2nm tolerance
Magnetic behaviour can be categorize in 3 regions in terms of the applied field: 1) low field region: linear approximation theory can be used.2) large field region: where full hysteresis models are applicable.3) transition region: ideal for magnetic hyperthermia: large SAR, less sensitive to size.
dM ( t)dt
=1τeff
( M 0( t)−M ( t) )
τeff=τN τB
τN+τBτB=
3ηVKbT
Brownian relaxation time Neel relaxation time
Magnetic particles will heat up
Cancer cells are more sensitive to heat
Possibility of developing a non-invasive cancer treatment
Biomedical limitation [2]:
f Hmax<6⋅107Oe / s
Apply an AC magnetic field
ΔU = Q − L
∫ M⃗⋅⃗dH
SAR (Specific absorption rate)=Energy
time⋅mass
NéelHAC
BrownHAC
Inmobilized (tumour tissue) Rotating (fluid)
• Experiment : SAR= cpΔT/Δt • Theory: SAR= HL∙f/Vt
M/M
S
H/HA
● The ability to predict particle heating is crucial for:● Controling the heating inside the human body.● Synthesizing the particles with optimal properties.
● Study of:● Intrinsic properties and their distribution (particle size,
anisotropy value, easy axis orientation).● Extrinsic properties (AC magnetic field amplitude, AC field
frequency).● The role of dipole interactions.● Environment effects (heat difuzion, Brownian rotation,
change of particle properties). (not considered here)
Hmax
=300 Oef=100 kHz
LRT
kMC
LRT
3
2
1
3 12
kMC calculations of SAR as a function of the mean particle size D assuming non-interacting system, in comparison with the predictions based on the RT (solid green lines). Considered are three values of mean anisotropy constant K: 3 × 105 erg/cm3 (circles, curve set 1 peaking at low D), 1.5 × 105 erg/cm3 (triangles, curve set 2 peaking in the intermediate D range), and 0.5 × 105 erg/cm3 (squares, curve set 3 saturating for large D).
Maximum SAR value (a) and peak position (b) as function packing fraction.
6) Conclusion We have developed a kinetic Monte-Carlo model of the underlying heating mechanisms associated with the
hyperthermia phenomenon used in cancer therapy. We show that the magnetic behaviour can be categorized in 3 regions in terms of the applied field:
1) the low field region where the linear response theory approximation, developed in previous studies, can be used, 2) the large field region where full hysteresis els are applicable and 3) an intermediate region where the transition between the two behaviour occurs and the conventional approaches
no longer apply. Magnetostatic interaction must be also considered.
Maximum SAR value (a) and the corresponding particle size (b) as function of anisotropy for: RT (black line), kMC non-interacting case (red circles) and kMC with interaction (green squares). In the fully hysteretic regime the SAR dependence on D does not present a peak, but is reaching a saturation value with increasing D.
● Magnetostatic interaction (random position of particle in a 3D configuration with volumetric packing fraction of 0.0, 0.05 and 0.10)
7) Reference[1] R. Rosensweig, “Heating magnetic fluid with alternating magnetic field,” Journal of Magnetism and Magnetic Materials, vol. 252, pp. 370–374, Nov. 2002.
[2] Hergt, R., & Dutz, S. (2007). Magnetic particle hyperthermia—biophysical limitations of a visionary tumour therapy. Journal of Magnetism and Magnetic Materials, 311(1), 187–192.
[3] E. Stoner and E. Wohlfarth, “A mechanism of magnetic hysteresis in heterogeneous alloys,” Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 240, no. 826, pp. 599–642, 1948.
[4] R. Chantrell, N. Walmsley, J. Gore, and M. Maylin, “Calculations of the susceptibility of interacting superparamagnetic particles,” Physical Review B, vol. 63,p. 024410, Dec. 2000.
[5] S. Ruta, R. Chantrell, and O. Hovorka, “Unified model of hyperthermia via hysteresis heating in systems of interacting magnetic nanoparticles.,” Sci. Rep., vol. 5, p. 9090, Jan. 2015.
LRT
KMC ε=0.0
KMC ε=0.1
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